Annals of Operations Research

, Volume 155, Issue 1, pp 79–105 | Cite as

The minimum shift design problem

  • Luca Di Gaspero
  • Johannes Gärtner
  • Guy Kortsarz
  • Nysret Musliu
  • Andrea Schaerf
  • Wolfgang Slany
Article

Abstract

The min-Shift Design problem (MSD) is an important scheduling problem that needs to be solved in many industrial contexts. The issue is to find a minimum number of shifts and the number of employees to be assigned to these shifts in order to minimize the deviation from workforce requirements.

Our research considers both theoretical and practical aspects of the min-Shift Design problem. This problem is closely related to the minimum edge-cost flow problem (MECF), a network flow variant that has many applications beyond shift scheduling. We show that MSD reduces to a special case of MECF and, exploiting this reduction, we prove a logarithmic hardness of approximation lower bound for MSD. On the basis of these results, we propose a hybrid heuristic for the problem, which relies on a greedy heuristic followed by a local search algorithm. The greedy part is based on the network flow analogy, and the local search algorithm makes use of multiple neighborhood relations.

An experimental analysis on structured random instances shows that the hybrid heuristic clearly outperforms our previous commercial implementation. Furthermore, it highlights the respective merits of the composing heuristics for different performance parameters.

Keywords

Workforce scheduling Hybrid algorithms Local search Greedy heuristics 

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References

  1. Aarts, E., & Lenstra, J. K. (Eds.). (1997). Local search in combinatorial optimization. New York: Wiley. Google Scholar
  2. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows, Englewood Cliffs: Prentice-Hall. Google Scholar
  3. Balakrishnan, N., & Wong, R. T. (1990). A network model for the rotating workforce scheduling problem. Networks, 20, 25–42. CrossRefGoogle Scholar
  4. Bar-Ilan, J., Kortsarz, G., & Peleg, D. (2001). Generalized submodular cover problems and applications. Theoretical Computer Science, 250(12), 179–200. CrossRefGoogle Scholar
  5. Bartholdi, J., Orlin, J., & Ratliff, H. (1980). Cyclic scheduling via integer programs with circular ones. Operations Research, 28, 110–118. Google Scholar
  6. Burke, E. K., De Causmaeker, P., Vanden Berghe, G., & Van Landeghem, H. (2004). The state of the art of nurse rostering. Journal of Scheduling, 7, 441–499. CrossRefGoogle Scholar
  7. Castro, J., & Nabona, N. (1996). An implementation of linear and nonlinear multicommodity network flows. European Journal of Operational Research, 92, 37–53. CrossRefGoogle Scholar
  8. Di Gaspero, L., & Schaerf, A. (2003). EasyLocal++: an object-oriented framework for flexible design of local search algorithms. Software Practice & Experience, 33(8), 733–765. CrossRefGoogle Scholar
  9. Di Gaspero, L., & Schaerf, A. (2006). Neighborhood portfolio approach for local search applied to timetabling problems. Journal of Mathematical Modeling and Algorithms, 5(1), 65–89. CrossRefGoogle Scholar
  10. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability—a guide to NP-completeness. San Francisco: W.H. Freeman. Google Scholar
  11. Gärtner, J., Musliu, N., & Slany, W. (2001). Rota: a research project on algorithms for workforce scheduling and shift design optimization. AI Communications: The European Journal on Artificial Intelligence, 14(2), 83–92. Google Scholar
  12. Glover, F., & Laguna, M. (1997). Tabu search. Dordrecht: Kluwer Academic. ISBN 0-7923-9965-X. Google Scholar
  13. Glover, F., & McMillan, C. (1986). The general employee scheduling problem: an integration of MS and AI. Computers & Operations Research, 13(5), 563–573. CrossRefGoogle Scholar
  14. Goldberg, A. V. (1997). An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms, 22, 1–29. CrossRefGoogle Scholar
  15. Hochbaum, D. (2000). Optimization over consecutive 1’s and circular 1’s constraints. Unpublished manuscript. Google Scholar
  16. Hoos, H. H., & Stützle, T. (1999). Towards a characterisation of the behaviour of stochastic local search algorithms for SAT. Artificial Intelligence, 112, 213–232. CrossRefGoogle Scholar
  17. Hoos, H. H., & Stützle, T. (2005). Stochastic local search: foundations and applications. San Francisco: Morgan Kaufmann. ISBN 1-55860-872-9. Google Scholar
  18. Jackson, W. K., Havens, W. S., & Dollard, H. (1997). Staff scheduling: a simple approach that worked (Technical report CMPT97-23). Intelligent Systems Lab, Centre for Systems Science, Simon Fraser University. Available at http://citeseer.nj.nec.com/101034.html.
  19. Johnson, D. S. (2002). A theoretician’s guide to the experimental analysis of algorithms. In M. H. Goldwasser, D. S. Johnson, & C. C. McGeoch (Eds.), Data structures, near neighbor searches, and methodology: fifth and sixth DIMACS implementation challenges (pp. 215–250). American Mathematical Society. URL http://www.research.att.com/~dsj/papers/experguide.ps.
  20. Krumke, S. O., Noltemeier, H., Schwarz, S., Wirth, H.-C., & Ravi, R. (1998). Flow improvement and network flows with fixed costs. In Proceedings of the international conference on operations research (OR-98), Zürich, Switzerland. Google Scholar
  21. Lau, H. C. (1996). On the complexity of manpower scheduling. Computers & Operations Research, 23(1), 93–102. CrossRefGoogle Scholar
  22. Musliu, N., Gärtner, J., & Slany, W. (2002). Efficient generation of rotating workforce schedules. Discrete Applied Mathematics, 118(12), 85–98. CrossRefGoogle Scholar
  23. Musliu, N., Schaerf, A., & Slany, W. (2004). Local search for shift design. European Journal of Operational Research, 153(1), 51–64. CrossRefGoogle Scholar
  24. Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: algorithms and complexity. Englewood Cliffs: Prentice-Hall. Google Scholar
  25. Raz, R., & Safra, S. (1997). A sub constant error probability low degree test, and a sub constant error probability PCP characterization of NP. In Proceedings of the 29th ACM symposium on theory of computing, El Paso (TX), USA (pp. 475–484). Google Scholar
  26. Tien, J. M., & Kamiyama, A. (1982). On manpower scheduling algorithms. SIAM Review, 24(3), 275–287. CrossRefGoogle Scholar
  27. Veinott, A. F., & Wagner, H. M. (1962). Optimal capacity scheduling: Parts I and II. Operation Research, 10, 518–547. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Luca Di Gaspero
    • 1
  • Johannes Gärtner
    • 2
  • Guy Kortsarz
    • 3
  • Nysret Musliu
    • 4
  • Andrea Schaerf
    • 1
  • Wolfgang Slany
    • 5
  1. 1.DIEGMUniversity of UdineUdineItaly
  2. 2.Ximes IncViennaAustria
  3. 3.Computer Science DepartmentRutgers UniversityCamdenUSA
  4. 4.Inst. for Information SystemsVienna University of TechnologyViennaAustria
  5. 5.Inst. for Software TechnologyGraz University of TechnologyGrazAustria

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